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Abstract
The experimental demonstration of measurement-device-independent quantum key distribution (MDI-QKD) has been widely demonstrated. Thus far, several experimental groups have implemented polarization encoding MDI-QKD but with manual polarization controllers, or polarization modulators that make circular polarization states unstable. Here, we apply an intrinsically stable polarization-modulated unit (PMU) to MDI-QKD so that Alice and Bob can modulate four BB84 polarization states, all of which can be kept stable from even the harsh environment. Moreover, our PMU can provide two operational polarization encoding modes suitable to different application scenarios. A proof-of-principle demonstration of MDI-QKD based on our PMU is implemented with an interference visibility of 46.6%, an average quantum bit error rate of 1.49% for the Z basis and the secure key rate of 4.25 × 10−6 bits per pulse. The proposed study is helpful for building polarization encoding MDI-QKD systems with better stability.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Quantum key distribution (QKD) protocols are designed to allow two parties, commonly known as Alice and Bob, to establish a string of shared secret keys, that are theoretically secure against even an all powerful adversary, Eve [1,2]. Combined with one time pad, such protocols can supply theoretically unconditional secure communication schemes [3]. In contrast to mathematical cryptography, QKD is based on physical techniques and its security has been proven [4,5]. Unfortunately, there is a large gap between ideal devices and actual imperfect setups, which have become the targets of many attacks [6–12]. Various schemes have been proposed so far to close the existing loopholes [13]. Given that most attacks, such as the time-shift attack [8], faked state attack [14,15], and detector blinding attack [9], have targeted Bob’s detection side, it is extremely practical to propose measurement-device-independent QKD (MDI-QKD) [16]. MDI-QKD is attractive not only because it can remove detector vulnerabilities, but also because of its practicality with respect to the current technology.
Currently, MDI-QKD is widely used as the most secure QKD system in experimental settings [17,18]. The schematic of a typical MDI-QKD is shown in Fig. 1. Two weak coherent pulses are independently sent and encoded by Alice and Bob. These pulses arrive and interfere at the beam splitter (BS) in the third node, usually known as Charlie, who does not need to be trusted. Then, Charlie performs partial Bell state measurements (BSMs) and publicly announces the results to Alice and Bob. Polarization encoding schemes [19–21] as well as phase and time-bin encoding schemes [22–24] are generally employed. Implementing MDI-QKD in a network setting over optical fibers, even though the phase and time-bin encoding scheme has the advantage of easily maintaining polarization indistinguishability, it requires expensive and bulky equipment for phase stabilization in the interferometers at the users’ sites, resulting in the expensive and complex setups of the end users. On the contrary, implementing polarization encoding is easier because it does not require to maintain interferometric stability that would be necessary for phase encoding. Although polarization encoding MDI-QKD requires expensive polarization compensation equipment to compensate for random birefringence fluctuations in optical fibers, in MDI protocols this implies sharing the equipment by several users in a star-like topology network. Thus, polarization encoding is more favorable than phase encoding in a MDI-QKD network. Additionally, since depolarization is practically negligible in free space, polarization encoding schemes are the better choices than phase and time-bin encoding schemes in free-space MDI-QKD.
Fig. 1. A typical MDI-QKD scheme. WCP, weak coherent pulse; IM, intensity modulator; BS, beam splitter; PBS, polarization beam splitter; D1H, D1V, D2H, and D2V, four single photon detectors; BSM, Bell state measurement.
In previously conducted polarization encoding MDI-QKD experiments, the polarization qubits have been encoded into polarization states using manual polarization controllers [19], or polarization modulators [20] which make circular polarization states unstable [25].
In this study, polarization encoding MDI-QKD has been proposed using intrinsically stable polarization-modulated units (PMUs) [26]. Two operational polarization encoding modes, namely the polarization-dependent mode and the polarization-independent mode, can be obtained by employing the PMU. Therefore, we can use the PMU in various situations. Also, using the PMU can achieve the four stable BB84 polarization states. The stability of polarization modulators has a great effect on MDI-QKD systems. On the one hand, the stability of polarization modulators will directly affect whether the long-term stable operation of MDI-QKD systems can be achieved. On the other hand, the instability of polarization modulators will lead to the deviation between the modulated polarization state and the ideal protocol polarization state, and thus cause the increase of the quantum bit error rate (QBER) and the decrease of the secure key rate. Our study brings the stable application of polarization encoding MDI-QKD in a network a step closer. Then, we apply the four-intensity decoy states [27] to MDI-QKD for defending attacks on an imperfect source, allowing us to obtain a lower bound for the final secure key rate.
2. System configuration of our MDI-QKD setup
Herein, a proof-of-principle demonstration of MDI-QKD has been provided by employing intrinsically stable PMUs [26]. The experimental setup is depicted in Fig. 2(a). Alice and Bob each use a continuous wave frequency-locked laser (Clarity-NLL-1550-LP, wavelength-1550 nm). The laser source is followed by a fast-axis locked polarization-maintaining circulator, which can well isolate the light reflecting from the fibers, thereby avoiding the influence of reflecting light on the performance of the laser. The laser source is modulated by a LiNbO3 intensity modulator (IM) to generate optical pulses with a temporal width of approximately 200 ps (FWHM) at a repetition rate of 25 MHz. For the stable operation of the IM, we apply a modulator bias controller (MBC-DG-BT-PD) (not shown herein) to the IM for bias control. A variable attenuator (ATT) is used to implement the four intensity decoy states protocol, which has a built-in mechanical shutter, allowing us to obtain a reliable estimation of the gain of the vacuum state. Four different mean photon numbers are obtained by independently adjusting the attenuation of the ATT. All the IMs and polarization modulators (Pol-Ms) are independently driven by electrical pulse generators (PGs). PGs and single photon detectors (SPDs) are synchronized by an electrical delay generator (DG) which is located inside Charlie’s station. Due to the limited number of available PGs driving the two Pol-Ms, we can only equally split the voltage generated by the PG3 (AVM-2-C-P) into two paths with two electrical attenuators (EATTs) to produce the required modulation voltage.
Fig. 2. (a) Schematic of the proposed MDI-QKD system. LD, laser diode; CIR, optical circulator; IM, intensity modulator; Pol-M, polarization modulator; ATT, attenuator; PC, manual polarization controller; QC, quantum channel; BS, beam splitter; PBS, polarization beam splitter; SPD, single photon detector; CC, coincidence counter; DG, delay generator; PG, electrical pulse generator; EATT, electrical attenuator. (b) Schematic of the Pol-M. PM, phase modulator; FM, Faraday mirror; OPM, optical power meter.
Notably, we should mention that phase randomization is naturally obtained because the time interval between consecutive pulses from Alice and Bob (40 ns) is much larger than the coherence time of the photons [28].
To conduct this experiment, the most important issue that needs to be resolved is to generate indistinguishable photons from two independent laser sources. To solve this problem, we use two frequency-locked lasers, whose wavelengths are independently locked to a molecular absorption line at approximately 1550.52 nm of an acetylene gas cell integrated in each laser by the manufacturer. The frequency linewidth of approximately 5 GHz after wave chopping is considerably larger than the frequency difference (within 10 MHz) between Alice’s laser and Bob’s laser. The arrival time of Alice’s pulses and Bob’s pulses at the BS can be independently adjusted by the DG with a resolution of 5 ps. Therefore, we can ensure that the two separately generated optical pulses possess sufficient overlap in both spectrum and time.
The key bits are encoded into the polarization states (+$45^{\circ }$, -$45^{\circ }$, right-hand circular, and left-hand circular) of the weak coherent pulses by the intrinsically stable PMU, which has been proposed in [26]. The schematic of the polarization modulator, which mainly comprises a four-port polarization beam splitter (PBS), a polarization-independent phase modulator (PM), and three Faraday mirrors (FMs), is shown in Fig. 2(b). The 1:99 BS and optical power meter (OPM) are used to monitor light intensity, ensuring that the polarization state of the pulses entering into the PBS is +$45^{\circ }$. In future implementations, this can be realized by automatic calibration with an electrical polarization controller (EPC).
The pulses are sent into the PMU through a single-mode fiber optic circulator (CIR) that allows the separation of the input and output optical pulses (see Fig. 2(b)). Each input optical pulse is divided into two orthogonal polarization components, $\left | {e_{x}} \right \rangle$ and $\left | {e_{y}} \right \rangle$, by the PBS. These two orthogonal polarization components then travel to the corresponding FMs, and superimpose on their way back from the FMs to the PBS. At this point, the application of appropriate, short voltage pulses to the PM can introduce a phase shift between the two components, achieving corresponding polarization modulation.
The detailed working principle of the PMU is elaborated in [26]. However, when the modulation method varies, the output polarization state of the PMU is rather different. As shown in Fig. 2(b), when all the apparatuses in the PMU are linked by single-mode optical fibers (SMFs), the input polarization state of the PM is arbitrary for its axes because of the birefringence of SMFs. However, no matter how harsh the external environment is and what the input polarization state of the PM becomes, as long as the same voltage is applied to the PM when one of the two components transmits through the PM twice, the PMU can modulate the desired stable polarization state. Then, we can name the modulation mode as the polarization-independent mode. On the other hand, as also shown in Fig. 2(b), when the fibers connecting port 1 of the PBS to the PM are polarization maintaining fibers (PMFs) and all the other fibers are SMFs, the input polarization state of the PM is aligned to either its slow axis or fast axis. Here, the PM needs to be modulated only once when one of the two components passes through the PM in either the forward-direction or reverse-direction. Then, the modulation mode is named as the polarization-dependent mode. Herein, we further describe the PMU’s principle, especially the two aforementioned modulation modes.
In common with [26], we calculate the polarization evolution of optical pulses using Dirac notation:
where $\left | {P_{out}} \right \rangle$ and $\left | {P_{in}} \right \rangle$ denote the polarization state of the output and input optical pulses, respectively, and $\hat P_{PMU}$ denotes the polarization transformation operator of the PMU.2.1 Polarization-independent mode
For convenience, we stipulate that the PBS transmits (reflects) the horizontally (vertically) polarized component $\left | {e_{x}} \right \rangle$ ($\left | {e_{y}} \right \rangle$). As shown in Fig. 2(b), when each optical pulse with +$45^{\circ }$ linear polarization state enters the PBS at port 0, the vertically polarized component $\left | {e_{y}} \right \rangle$ moves into port 1, whereas the horizontally polarized component $\left | {e_{x}} \right \rangle$ moves into port 2. We consider the component $\left | {e_{y}} \right \rangle$ with voltage pulses being applied to the PM, whereas the component $\left | {e_{x}} \right \rangle$ enters the PM without the application of control voltage pulses, as an example to illustrate the modulation process. When the component $\left | {e_{y}} \right \rangle$ arrives at the PM after traveling through the single mode fibers, because of birefringence, the polarization state of the component $\left | {e_{y}} \right \rangle$ transforms as follows:
The operator of the Faraday mirror (FM) can be expressed as follows:
As can be seen from Eq. (2) to Eq. (8), the polarization state of the initial component $\left | {e_{y}} \right \rangle$ tranforms from $\left | {\psi _{1}}\right \rangle$ to $\left | {\psi _{6}}\right \rangle$ when the component reaches the PBS on its way back from FMs (FM1, FM2 and FM3) to the PBS. And thus the initial component $\left | {e_{y}} \right \rangle$ has a phase shift of $(\varphi _{e} + \varphi _{o})$ when voltage pulses are applied to the PM twice, and its polarization state transforms to $\left | {e_{x}} \right \rangle$. Meanwhile, the component $\left | {e_{x}} \right \rangle$ enters the PM without applying the voltage pulses, and thus, no additional phase is generated. When the initial component $\left | {e_{x}} \right \rangle$ reaches the PBS on its way back from FMs (FM1, FM2 and FM3) to the PBS, according to Eq. (4), the polarization state of the initial component $\left | {e_{x}} \right \rangle$ tranforms as follows:
Finally, both the orthogonal components simultaneously arrive, and superimpose at the PBS. If the input polarization state is +$45^{\circ }$ ($\left | {P_{in}} \right \rangle = (\left | {e_{x}} \right \rangle +\left | {e_{y}} \right \rangle )/{\sqrt {2 }}$), we can obtain the polarization state of the output optical pulses according to Eqs. (8) and (9).Similarly, the modulation process is the same for the component $\left | {e_{x}} \right \rangle$ with voltage pulses being applied to the PM, whereas the component $\left | {e_{y}} \right \rangle$ enters the PM without the application of control voltage pulses.
2.2 Polarization-dependent mode
In the polarization-dependent mode, we consider the component $\left | {e_{y}} \right \rangle$ with voltage pulses being applied to the PM, whereas the component $\left | {e_{x}} \right \rangle$ enters the PM without the application of control voltage pulses, as an example to illustrate the modulation process. As also shown in Fig. 2(b), when the component $\left | {e_{y}} \right \rangle$ reaches the PM after traveling through the PMFs, the polarization state of the component $\left | {e_{y}} \right \rangle$ is still $\left | {e_{y}} \right \rangle$. At this moment, voltage pulses are applied to the PM, and the polarization state of the component $\left | {e_{y}} \right \rangle$ is modified as follows:
The influence of birefringence can be ignored because the go-and-return configuration automatically compensates for the birefringence and phase drift. When the two orthogonal components finally simultaneously arrive at the PBS on their way back from FMs (FM1, FM2 and FM3), the polarization state of the component $\left | {e_{y}} \right \rangle$ transforms into $\left | {e_{x}}\right \rangle$ with a phase shift of $\varphi _{o}$, whereas the polarization state of the component $\left | {e_{x}} \right \rangle$ turns into $\left | {e_{y}} \right \rangle$. The two orthogonal polarization components superpose at the PBS. If the input polarization state is +$45^{\circ }$ ($\left | {P_{in}} \right \rangle = (\left | {e_{x}} \right \rangle +\left | {e_{y}} \right \rangle )/{\sqrt {2 }}$), and the polarization state of the output optical pulses is obtained.2.3 Summary for the PMU
According to Eqs. (10) and (12), when four phase shifts of $\varphi _{o}, (\varphi _{e} + \varphi _{o})= 0, \pi , \pi /2, 3\pi /2$ are generated, corresponding to four different voltages of 0, $V_{\pi }$, $V_{\pi }/2$, $3V_{\pi }/2$, the output modulated polarization states are +$45^{\circ }$, -$45^{\circ }$, left-hand circular (L), right-hand circular (R), respectively. The first two polarization states belong to the diagonal (Z) basis and the latter two polarization states belong to the circular (X) basis.
When the PMU operates in the polarization-independent mode, we apply the control voltage pulses to the PM twice. Therefore, the voltage required to modulate the corresponding polarization state is relatively small in comparison with that required in the the other mode. When the PMU operates in the polarization-dependent mode, it is easier to control in the timing and to achieve high-speed modulation. Moreover, for the polarization-independent mode, PMFs are not needed and all the devices in the PMU are linked by SMFs so that the PMU can be applied in poor working conditions, such as those in the outdoor military environment. For the polarization-dependent mode, PMFs are needed to connect port 1 of the PBS to the PM. Since PMFs can not keep the polarization state stable in harsh environment, we can only use the PMU in relatively stable situations, such as indoor conditions. To sum up, the operational mode of the PMU can be chosen flexibly according to the actual situation of the system.
The polarization modulator proposed in [25] exhibits stability for the diagonal basis, but instability for the circular polarization states because of the existence of the phase introduced by PMF [25]. The PMF exacerbates changes in polarization states which are not aligned to its axes. We verified experimentally that the modulated circular polarization states with the modulator proposed in [25] are very unstable and change very quickly within 5 minutes, as the theory indicates. The stability data on the polarization circular states even within 5 minutes can not be obtained.
Fortunately, the modulated polarization states with our PMU are theoretically intrinsically stable. Also, a polarization extinction ratio (PER) of about 30 dB was maintained for more than eight hours without any adjustments with the PMU, which is shown in [26].
Herein, Fig. 3(a) and Fig. 3(b) show the variation of PER over time of the left-hand circular polarization state (the modulated voltage of $V_{\pi }/2$ is required) and the right-hand circular polarization state (the modulated voltage of $3V_{\pi }/2$ is required) with our PMU, respectively. The optical pulses exiting the PMU are split by a PBS and the photon counts are measured at the two outputs. +$45^{\circ }$ (the modulated voltage of 0 is required) and -$45^{\circ }$ (the modulated voltage of $V_{\pi }$ is required) polarization states are aligned to the axes of the PBS. The PER is stipulated as the ratio between the photon counts in the two outputs. Therefore, Fig. 3(a) and Fig. 3(b) showing the variation of PER over time can reflect the stability of the modulated circular polarization states. We convert the above PER to the circular basis, and obtain a PER of exceeding 30.6 dB within 30 minutes. These experimental data within 30 minutes have fully proved that the stability of our PMU is better than other modulators. When using our PMU as the source of the photon signal by both Alice’s and Bob’s stations in MDI-QKD systems, based on the decoding principle, the following formula can be derived to calculate the QBER caused by the circular polarization state deviation.
Fig. 3. (a) the variation of PER over time of the left-hand circular polarization state. (b) the variation of PER over time of the right-hand circular polarization state.
3. Results and discussion
In this proof-of-principle experiment, as shown in Fig. 2(b), for the sake of simplicity, Alice and Bob sent their pulses to Charlie without going through real fiber spools. If such spools were used, we would need to add automatic polarization control systems to compensate for polarization changes caused by fiber spools [19], and timing-calibration technique to ensure precise and stable overlap in the arrival time at Charlie’s BS [23,24]. The results can be obtained by employing variable attenuators to reproduce the attenuation of single mode fibers (about 0.2 dB/km) [21]. In our system, the absorption value simulated for the channel does not affect the feasibility of the system, and the absorption value we apply is approximately 0. In future work, we will apply real fiber spools. As shown in Fig. 2(b), Charlie’s setup comprises a 50:50 BS, a PBS, two SPDs and a coincidence counter (CC). The SPDs employed are commercial InGaAs/InP detectors working in gated mode, with 300-ps-gate windows, average dark count probability per gate of $1.1\times 10^{-6}$, 100 ns dead time and detection efficiency of approximately 10$\%$. We choose to detect photons only at the outputs of one PBS owing to the lack of SPDs. When these two detectors click within a coincidence window of 800 ps, we obtain a coincidence count measured by the coincidence counter. A coincidence count corresponds to a successful projection into the triplet state ($\left | {\psi ^{+}}\right \rangle = (\left | {HV} \right \rangle + \left | {VH} \right \rangle )/\sqrt {2}$). We employ manual polarization controllers (PC3 and PC4) to make Alice and Bob share a polarization reference frame.
The feasibility of the experiment is dependent on obtaining a high Hong-Ou-Mandel (HOM) interference visibility. We set the two separate pulses from Alice and Bob to be indistinguishable from the viewpoint of arrival time and spectrum. Then, the applied control voltage on the PM of the PMU inside Alice’s setup is maintained at 0 V. Afterwards, we change the applied control voltage on Bob’s PM from 0 V to approximately 7.5 V. For these values, we can obtain an interference visibility of 46.6$\%$, which is close to the theoretical maximum of 50$\%$.
We implement the four-intensity decoy-state MDI-QKD protocol to estimate the lower bound of the final secure key rate. Alice and Bob each produce only one intensity $\mu _z= 0.6$ in the Z basis with a probability of 0.827, and three different intensities 0, $\mu _x= 0.05$, and $\mu _y= 0.19$ in the X basis with probabilities of 0.02, 0.128 and 0.025, respectively. The Z basis is used to distill the key bits, whereas the X basis is used to test the noise and error rate. The average number of photons per pulse and the probability of each intensity are selected according to [21,24]. The experimental values of gains and quantum bit error rates (QBERs) are listed in Table 1 and Table 2, respectively. Then, we can use the formula [16] to estimate the secure key rate R.
Table 1. Experimental values of the gain $Q ^{\mu _{z},\mu _{z}}_{Z}$ and quantum bit error rate (QBER) $E ^{\mu _{z},\mu _{z}}_{Z}$ in the diagonal (Z) basis.
Table 2. Experimental values of gains $Q ^{\mu _{i},\mu _{j}}_{X}$ and QBERs $E ^{\mu _{i},\mu _{j}}_{X}$ in the circular (X) basis. Since there is only one electrical pulse generator (PG3) driving two Pol-Ms, when Alice modulates the right-hand circular polarization state (the voltage of $3V_{\pi }/2$ is required), Bob can only simultaneously generate a voltage close to $3V_{\pi }/2$, not exactly $3V_{\pi }/2$, and vice versa. As a result, we obtain fewer coincidence counts than those obtained by simultaneous modulation of the left-hand circular polarization state, resulting in slightly higher QBERs.
Table 3. Parameters used to extract the lower bound of the secure key rate.
We can improve the secure key rate by increasing the repetition rate of the system and detection efficiency of SPDs. We can also perform a numerical simulation and parameter optimization to optimize the performance. In addition, if four detectors are employed instead of two, the secure key rate can be increased by at least four times. Moreover, if we use two electrical PGs rather than one, the QBERs in the circular basis can be lower.
4. Conclusion
In conclusion, a proof-of-principle demonstration of polarization encoding MDI-QKD based on the intrinsically stable PMUs which can modulate four intrinsically stable BB84 polarization states is performed. The feasibility of the presented system has been manifested with an interference visibility of 46.6$\%$, an average QBER of 1.49$\%$ for the Z basis, and the secure key rate of $4.25\times 10^{-6}$ bits per pulse. Besides, we further describe the modulation process of the PMU and prove that the PMU can supply two operational polarization encoding modes, which can be valuable for achieving stable polarization modulation in various situations. Our work can improve polarization encoding MDI-QKD implementations by improving polarization modulation stability, and is helpful for building a polarization encoding MDI-QKD network capable of stable operation in practical applications.
Funding
National Natural Science Foundation of China (61771205); Natural Science Foundation of Guangdong Province (2015A030313388); Science and Technology Planning Project of Guangdong Province (2015B010128012, 2017KZ010101).
Acknowledgments
The authors thank Prof. Wei Chen, Prof. Shi-hai Sun, and Prof. Shuang Wang for their useful discussions.
Disclosures
The authors declare no conflicts of interest.
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